Linear imperfections and correction - Yannis Papaphilippou - CERN

Linear imperfections and correction Imperfections, USPAS, January 2008

Y. Papaphilippou CERN United States Particle Accelerator School, University of California - Santa-Cruz, Santa Rosa, CA 14th – 18th January 2008

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Linear Imperfections and correction Steering error and closed orbit distortion

Imperfections, USPAS, January 2008

Gradient error and beta beating correction Linear coupling and correction Chromaticity 2

Beam stability in storage rings  Beam orbit stability very critical especially for the stability of the synchrotron light spot in the beam lines  Consequences of orbit distortion 

Miss-steering of photon beams, modification of the dispersion function, resonance excitation, aperture limitations, lifetime reduction, coupling of beam motions, modulation of lattice functions, poor injection efficiency

 Long term Causes (Years - months) Imperfections, USPAS, January 2008



Ground settling, season changes, diffusion, medium - Days/Hours, sun and moon, day-night variations (thermal), rivers, rain, water table, wind, synchrotron radiation, refills and start-up, sensor motion, drift of electronics, local machinery, filling patterns

 Short (Minutes/Seconds) 

Ground vibrations, power supplies, injectors, insertion devices, air conditioning, refrigerators/compressors,water cooling, beam instabilities in general 3

Closed orbit distortion  Causes  Dipole

field errors  Dipole misalignments  Quadrupole misalignments  Consider the displacement of a particle δx from the ideal orbit . The vertical field is


quadrupole dipole

 Remark: Dispersion creates a closed orbit distortion for off-momentum particles  Effect of orbit errors in any multi-pole magnet  Feed-down

2(n+1)-pole

2n-pole

2(n-1)-pole

dipole4

Effect of single dipole kick • Introduce Floquet variables


• • • •

The Hill’s equations are written The solutions are the ones of an harmonic oscillator Consider a single dipole kick at φ=π Then and and in the old coordinates

Maximum distortion amplitude

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Example: Orbit distortion in ESRF storage ring


 In the ESRF storage ring, the beta function is 1.5m in the dipoles and 30m in the quadrupoles.  Consider dipole error of δy’=1mrad  The horizontal tune is 36.44  Maximum orbit distortion in dipoles

 For quadrupole displacement with 1mm, the distortion is  Magnet alignment is critical

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Closed orbit distortion

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Many orbit errors’ effect  Consider random distribution of errors in N magnets  The expectation value is given by

 Example: Imperfections, USPAS, January 2008

 In

the ESRF storage ring, there are 64 dipoles  The expectation value of the orbit distortion in the dipoles is 4mm, considering a 1mrad dipole error  The expection value for the orbit distortion created by the 112 focusing quadrupoles, considering an equivalent 0.1mrad dipole error from misalignments (~ 0.1mm for 1m long quads of 10T/m gradient), is 11mm! 8

Transport of orbit distortion due to dipole kick • Consider a transport matrix between positions 1 and 2

• The transport of transverse coordinates is written as


• Consider a single dipole kick at position 1 • Then, the first equation may be rewritten • Replacing the coefficient from the general betatron matrix

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Correcting the orbit distortion


 Place horizontal and vertical dipole correctors close to the corresponding quads  Simulate (random distribution of errors) or measure orbit in Beam position monitors (downstream of the correctors)  Minimize orbit distortion with several methods  Globally  Harmonic , which minimizes components of the orbit frequency response after a Fourier analysis  Most efficient corrector (MICADO), finding the most efficient corrector for minimizing the rms orbit  Least square fitting  Locally

 Sliding Bumps  Singular Value Decomposition (SVD) 10

Orbit bumps


 2-bump: Only good for phase advance equal π between correctors  Sensitive to lattice and BPM errors  Large number of correctors

 3-bump: works for any lattice  Need large number of correctors  No control of angles

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Singular Value Decomposition example

M. Boege, CAS 2003

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Orbit feedback


Closed

orbit stabilization performed using slow and fast orbit feedback system. Slow operates every a few seconds (30 for the ESRF) and uses complete set of BPMs (~200 at the ESRF) for both planes Efficient in correcting distortion due to current decay in magnets or other slow processes Fast orbit correction system operates in a wide frequency range  (.1Hz to 150Hz for the ESRF) correcting distortions induced by quadrupole and girder vibrations. Local feedback systems used to damp horizontal orbit distortion in ID straight sections, especially the ones were beam spot stabilization is critical

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Gradient error and optics distortion  Key issue for the performance -> super-periodicity preservation -> only structural resonances excited  Broken super-periodicity -> excitations of all resonances  Causes


 Errors

in quadrupole strengths (random and systematic)  Injection elements  Higher-order multi-pole magnets and errors

 Observables  Tune-shift  Beta-beating  Excitation

of integer and half integer resonances 14

Gradient error  Consider the transfer matrix for one turn


 Consider a gradient error in a quad. In thin element approximation the quad matrix with and without error are

 The new 1-turn matrix is which yields

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Gradient error and tune-shift  Consider a new matrix after 1 turn with a new tune


 The traces of the two matrices describing the 1-turn should be equal which gives  Developing the left hand side

 and finally  For a quadrupole of finite length, we have 16

Gradient error and beta distortion  Consider the unperturbed transfer matrix for one turn

 Introduce a gradient perturbation between the two matrices


 Recall that term as

and write the perturbed

 On the other hand and  Equating the two terms and integrating through the quad 17


Example: Gradient error in the ESRF storage ring  Consider 128 focusing arc quads in the SNS ring with 0.1% gradient error. In this location =30m. The length of the quads is around 1m  The tune-shift is

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Random gradient error distribution  For a random distribution of errors the beta


variation (beating) is

 Optics functions beating >10% by putting random errors (0.1% of the gradient) in high dispersion quads of the ESRF storage ring  Justifies the choice of quadrupole corrector strength

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Gradient error correction  Windings on the core of the quadrupoles or individual quadrupole correctors (TRIM)  Simulation by introducing random distribution of quadrupole errors  Compute the tune-shift and the optics function beta distortion  Move working point close to integer and half integer resonance  Minimize beta wave or quadrupole resonance width with TRIM windings  To correct certain resonance harmonics N, strings should be powered accordingly  Individual powering of TRIM windings can provide flexibility and beam based alignment of BPM 20

Linear coupling  Betatron motion is coupled in the presence of skew quadrupoles


 The field is and Hill’s equations are coupled  Motion still linear with two new eigen-mode tunes, which are always split. In the case of a thin quad:  Coupling coefficients  As motion is coupled, vertical dispersion and optics function distortion appears  Causes:  Random rolls in quadrupoles  Skew quadrupole errors 21  Off-sets in sextupoles

Vertical emittance  Ideally the vertical dispersion is zero and the vertical dispersive emittance is zero, so the vertical emittance  Real Lattice: Errors as sources of vertical emittance Vertical dipoles (a1) and skew quadrupoles (a2)  Dipole and quadrupole rolls  Quadrupole and Sextupole vertical displacement 


 Result to vertical dispersion and linear coupling needing orbit correctors and skew quadrupoles for suppression  Emittance ratio  Coupling corrected when coupling coefficient drops to 10−3 level  Brightness proportional to the inverse of the coupling coefficient only for hard X-rays (diffraction limitation)  Touschek lifetime proportional to

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Linear coupling correction strategy Introduce skew quadrupole correctors Simulation by introducing random distribution of quadrupole errors Correct globally/locally coupling coefficient (or resonance driving term) Correct optics distortion (especially vertical dispersion) Move working point close to coupling resonances and repeat 23

Example: Coupling correction for the ESRF ring


 Local decoupling using 16 skew quadrupole correctors and coupled response matrix reconstruction  Achieved correction of below 0.25% reaching vertical emittance of below 10pm

R. Nagaoka, EPAC 2000 24


Chromaticity  Linear equations of motion depend on the energy (term proportional to dispersion)  Chromaticity is defined as:  Recall that the gradient is  This leads to dependence of tunes and optics function on energy  For a linear lattice the tune shift is:  So the natural chromaticity is: 25

Example: Chromaticity in the SNS ring


 In the SNS ring, the natural chromaticity is –7.  Consider that momentum spread  The tune-shift for off-momentum particles is

 In order to correct chromaticity introduce particles which can focus off-momentum particle Sextupoles 26

Chromaticity from sextupoles  The sextupole field component in the x-plane is:  In an area with non-zero dispersion  Than the field is quadrupole

dipole


 Sextupoles introduce an equivalent focusing correction  The sextupole induced chromaticity is  The total chromaticity is the sum of the natural and sextupole induced chromaticity 27

Chromaticity correction


 Introduce sextupoles in high-dispersion areas (not easy to find)  Tune them to achieve desired chromaticity  Two families are able to control horizontal and vertical chromaticity  Sextupoles introduce non-linear fields (chaotic motion)  Sextupoles introduce tune-shift with amplitude  Example: The ESRF ring has natural chromaticity of -130  Placing 32 sextupoles of length 0.4m in locations where β=30m, and the dispersion D=0.3m  For getting 0 chromaticity, their strength should be 

or a gradient of 280 T/m2 28

Eddy current sextupole component Sextupole component due to Eddy currents in an elliptic vacuum chamber of a pulsing dipole

with


Taking into account

with

we get 29

Measurements vs Theory Example: ESRF booster chromaticity Booster Chromaticity without correction 5


Chromaticity

-5

-15

-25 Horizontal (measurement) Vertical (measurement)

-35

Horizontal (theory) Vertical (theory)

-45 0

20

40

60

Time (ms)

80

100

120 30


Two vs. four families for chromaticity correction

 Two families of sextupoles not enough for correcting off-momentum optics functions’ distortion and second order chromaticity  Solutions:  Place sextupoles accordingly to eliminate second order effects (difficult)  Use more families (4 in the case of of the SNS ring)  Large optics function distortion for momentum spreads of ±0.7%,when using only two families of sextupoles 31  Absolute correction of optics beating with four families

References


O. Bruning, Accelerators, Summer student courses, 2005. H. Wiedemann, Particle Accelerator Physics I, Springer, 1999. K.Wille, The physics of Particle Accelerators, Oxford University Press, 2000.

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